Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets
Algebraic Geometry
2007-05-23 v1 Symbolic Computation
Algebraic Topology
Abstract
In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results.
Keywords
Cite
@article{arxiv.math/0603248,
title = {Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets},
author = {Saugata Basu and Richard Pollack and Marie-Francoise Roy},
journal= {arXiv preprint arXiv:math/0603248},
year = {2007}
}